From Metric to Connection. Actions for Gravity, with Generalizations

Abstract: The search for a theory of quantum gravity has for a long time been almost fruitless. A few years ago, however, Ashtekar found a reformulation of Hamiltonian gravity, which thereafter has given rise to a new promising quantization project; the canonical Dirac quantization of Einstein gravity in terms of Ashtekar's new variables. This project has already given interesting results, although many important ingredients are still missing before we can say that the quantization has been successful.

Related to the classical Ashtekar Hamiltonian, there have been discoveries regarding new classical actions for gravity in (2+1)- and (3+1)-dimensions, and also generalizations of Einstein's theory of gravity. In the first type of generalization, one introduces infinitely many new parameters, similar to the conventional Einstein cosmological constant, into the theory. These generalizations are called "neighbours of Einstein's theory" or "cosmological constants generalizations", and the theory has the same number of degrees of freedom, per point in spacetime, as the conventional Einstein theory. The second type is a gauge group generalization of Ashtekar's Hamiltonian, and this theory has the correct number of degrees of freedom to function as a theory for a unification of gravity and Yang-Mills theory. In both types of generalizations, there are still important problems that are unresolved: e.g. the reality conditions, the metric-signature condition, the interpretation, etc.

In this thesis, I will try to clarify the relations between all these new actions, and also give a short introduction to the new generalizations. This is done in the first part of the thesis, and in the second part, papers I-VIII are reprinted.

Papers I, II and IV treat the inclusion of the cosmological constant and matter couplings into the pure spin-connection CDJ-formulation, which is related to the Ashtekar Hamiltonian through a Legendre transform. In paper III, one new neighbour of Einstein's theory is analyzed. We calculate the static and spherically symmetric solution to this theory, and use the weak field approximation to set experimental bounds on this new cosmological constant. In paper V, I scrutinize an old uniqueness proof for the Einstein theory, and it is shown that the proof only holds provided the metric has a simple relation to the phase space coordinate. The (2+1)-dimensional pure spin-connection CDJ-Lagrangian is found in paper VI, where I also show that the new cosmological constants generalization, that exists in (3+1)-dimensions, has no direct counterpart in (2+1)-dimensions. Paper VII shows how to find a gauge group generalization of the Ashtekar Hamiltonian. This generalization may lead to a unification of gravity and Yang-Mills theory. And, finally, in paper VIII the corresponding gauge group generalization of Ashtekar's variables in (2+1)-dimensions is found, and it is shown that this generalization really has an interpretation as gravity coupled to Yang-Mills theory, in (2+1)-dimensions.